DYNAMICS OF A RIGID BODY. 97 



If the condition U^ = o be fulfilled, then the two 

 screws TJ, , and the two screws on the cylindroid (TJ, ), 

 which belong to the complex UQ = o, are parallel to the 

 four rays of an harmonic pencil ( 89). 



We can now deduce a result of some importance. If 

 we regard the screw r\ as being given, then the screw 

 must belong to a screw complex of the fifth order and 

 first degree, which is defined by : 



This complex may be constructed in the following 

 manner : Draw any cylindroid through TJ, find on this 

 the two screws which belong to U e = o, then a fourth 

 screw ? can be determined by the condition that the set 

 shall be parallel to the rays of an harmonic pencil. 

 The same process repeated for four other cylindroids 

 through 77, will give five screws, by which the screw 

 complex to which belongs is determined. 



It will be observed that in the determination of the 

 screw complex U^ = o, where TJ [is given, no occasion 

 has arisen for making mention of the screws of reference 

 to which the co-ordinates are referred. If, further, it be 

 observed, that all the screws of the complex 6^ = are 

 reciprocal to that one screw of which the co-ordinates 

 are proportional to 



p\ \ dkji / ' ' A ^ 



we have the following theorem : 



If Ue - o denote a screw complex of the fifth order 

 and second degree, then to every screw r\ corresponds with 

 respect to the screw complex, a polar screw, whose co-ordi- 

 nates are proportional to 



r "/. 



H 



